The cognitive effects of a mathematics in-service workshop on elementary school teachers

The cognitive effects of a mathematics in-service workshop on elementary school teachersAuthor(s): WILLIAM M. BART and ROBERT E. ORTONSource: Instructional Science, Vol. 20, No. 4 (1991), pp. 267-288Published by: SpringerStable URL: http://www.jstor.org/stable/23369963 .

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Instructional Science 20: 267-288 (1991) 267 © Kluwer Academic Publishers, Dordrecht - Printed in the Netherlands

The cognitive effects of a mathematics in-service workshop on

elementary school teachers

WILLIAM M. BART & ROBERT E. ORTON

College of Education, University of Minnesota, Minneapolis, MN 54455, U.S.A.

Abstract. Siegler's rule assessment methodology was used to investigate the cognitive effects of a 32

hour mathematics in-service workshop on 28 elementary school teachers. This paper reports on

attempts (a) to assess the levels of cognitive understanding of the basic concepts of proportion, proba

bility, and correlation among elementary teachers, and (b) to change teachers' levels in understanding these concepts by a method of "direct instruction". Significant improvements were noted in the levels

of cognitive development associated with the concept of proportion. Though nonsignificant improve ments were noted in the concepts of probability and correlation, all the teachers were assigned to the

highest cognitive rule associated with the probability concept at the end of the workshop.

Introduction

Mathematics instruction is a major component in elementary education. If ele

mentary school teachers are to be successful in teaching mathematics, then they

need to understand the basic mathematical content that they will be teaching.

Unfortunately, many elementary teachers are deficient in the mathematical

knowledge that they teach their students (Lacampagne et al., 1988; Post et al.,

1988). From a sample of over 200 elementary teachers in a large Midwestern city,

a little over 50% could divide 1/3 by 3, and less than 50% could divide 3 by 4/3

(Post et al., 1988). A remediation program to retrain less knowledgeable teachers in these "basic

skills" might be one solution to this problem. However, mathematics educators

argue that remediation programs that merely reteach the basic skills do not

address the underlying problem, which is a faulty understanding of the concepts and principles behind these basic skills (NCSM, 1978; NCTM, 1980). Ideally, a

remediation program would aim at cognitive training in the concepts underlying

elementary school mathematics.

One set of fundamental concepts underlying elementary school mathematics

are those of proportion, probability and correlation that are identified in the work

of Piaget (Inhelder and Piaget, 1958; Siegler, 1981). Although the concepts are

fundamental, they are not simple one-variable concepts such as height and

weight. These concepts entail the coordination of several variables and thus are

sophisticated concepts.



268

In the case of proportion, the concept is applicable to any situation involving

variables a, b, c, and d if the variables satisfy an equality of two ratios such as

a/b = c/d. An example of such a situation involves a balance scale with two

weights in which (a) the weight on the left side of the fulcrum weighs W, ounces

and is Dj inches from the fulcrum, (b) the weight on the right side of the fulcrum

weighs W2 ounces and is D2 inches from the fulcrum, and (c) the scale is level. If

these variables Wj, Dj, W2, and D2 satisfy the proportional relationship

Wj/W2= D2/Dj, then the concept of proportion is applicable to this situation. A

proportional relationship allows one to make correct predictions regarding the

value of one of the variables if one knows the values of the other three variables.

Children with inadequate understanding of these concepts will often ignore one

of the relevant variables and make judgments based on a proper subset of the rele

vant variables. Thus, such a child will say that a balance scale is level if a 5 ounce

weight is placed 2 inches from the fulcrum on the left side of the scale and a

5 ounce weight is placed 4 inches from the fulcrum on the right side, because the

two weights are equal. To such an individual, the distance of either weight from

the fulcrum is irrelevant.

As children develop more sophisticated understanding of multiple variable

concepts such as proportion and probability, they learn to encode all of the rele

vant variables and they learn how the variables are to be coordinated. Inhelder

and Piaget contended that adolescents and adults will have well-developed con

cepts of proportionality, probability and correlation only after they can coordinate

several variables into an integrated system.

Inhelder and Piaget used clinical interviews to assess levels of understanding

of adolescents regarding these three fundamental concepts. Though the interviews

provide some corroboration for their theory, that theory lacks precision in terms

of differentiating among levels of understanding of the concepts. That precision can be garnered by using a detailed information processing model of cognitive

processing. A clear analysis of an intelligence task was provided by Siegler (1976) regard

ing the Inhelder-Piaget balance task. He used information processing methods to

examine the concept of proportionality. Siegler posited that there will be develop mental differences with respect to the balance task, because children at different

ages will demonstrate different levels of understanding of the concept by employ

ing different cognitive rules in their responses to the task. For his task items,

Siegler posited four cognitive rules, each of which indicates a different level of

understanding of the proportionality concept. These cognitive rules are explained in greater detail later in this article.

Siegler's rule-based assessment method, though still debatable, makes possible

precise testing of Inhelder and Piaget's ideas regarding the growth of

sophisticated logical ideas in children, adolescents and adults. This attribute of

"testability" is one of the hallmarks of scientific theorizing (Popper, 1963). The



269

results described in this article also indicate that Siegler's work is useful in

assessing the cognitive effects of direct instruction with adult learners. This

attribute of "fruitfulness" (or explanatory power) is another hallmark of scientific

theorizing (Lakatos, 1970). This paper reports on attempts (a) to assess the levels of cognitive understand

ing of the basic concepts of proportion, probability and correlation among elementary teachers, and (b) to change teachers' levels in understanding these

concepts by a method of "direct instruction".

Siegler' s rule assessment methodology

Robert Siegler (1981) fashioned two sets of 24 tasks each of which assessed

understanding of the concepts of proportion and probability, respectively. Each task provided three choices. For example, in any of the probability tasks, the sub

ject would be shown two sets of marbles. In each set (set A or set B), there would

be some white marbles and some black marbles. The subject would be asked

which set of marbles provides the best chances of picking a black marble with the

three choices being (a) set A, (b) set B, and (c) both sets provide the same chance.

By examining the response patterns to the tasks, Siegler was able to classify sub

jects according to four well-defined cognitive rules (or problem solving

strategies) which reflected levels of cognitive development regarding those con

cepts. The rules were termed rules 1, 2, 3, and 4, with rule 1 indicating the lowest level of cognitive development and rule 4 indicating the highest level of cognitive

development.

For each of the three mathematical concepts, the four well-defined cognitive

rules were similar in terms of their relative complexities; they were different in

terms of the component stimulus features being addressed and in terms of the

methods of combining those features. Each of the concepts assessed entailed two

variables, a dominant variable and a subordinate variable. For example, the

proportionality concept assessed by Siegler's balance task items involved the

dominant variable of weight (i.e., the weight on one side of the balance) and the

subordinate variable of distance (i.e., the distance from the fulcrum of the weight on one side of the balance). For each of the four rules associated with any of the

concepts, the values of either the dominant variable or both variables are consid

ered. Rule 1 responses indicated that the subjects attended only to the dominant

variable of the problem situation, whereas rule 4 responses indicated that the sub

jects attended to both dominant and subordinate variables at the same time.

One way to explain these rules is to use flowcharts. For example, as shown in

Figure la, according to rule 1, the subject chooses the side (or set) which has the

greater value for the dominant variable (DL or DR); if the two sides (or sets) have

equal values for the dominant variable, then the subject selects the third choice,

i.e., the choice indicating equivalence between the two sides (or sets).



270

According to rule 2 (depicted in Figure lb), the subject chooses the side with

the greater subordinate variable value (SL or SR) only if the dominant variable

values are equal; otherwise, the subject follows rule 1. According to rule 3

(Figure lc), the subject chooses the side (or set) with the greater dominant varia ble value if that side (or set) also has the greater subordinate value or guesses if

the side (or set) with the greater dominant variable value is the side (or set) with the smaller subordinate variable value; otherwise, the subject follows rule 2.

According to rule 4 (Figure Id), the subject employs an algorithm that generates the correct answer if the side (or set) with the greater dominant variable value has

the smaller subordinate variable value; otherwise, the subject follows rule 3.

Awang (1984) developed paper-and-pencil versions of the Siegler task sets and

fashioned a comparable test for the concept of correlation composed of 24 items.

Awang used the tests to examine the levels of understanding of the concepts of

probability, proportionality and correlation among American and Malaysian

college students. Examples of the Awang items are provided in Figures 2-4. The

tasks (test items) associated with the concepts of proportionality, probability and

correlation will be referred to as the balance beam, marbles, and fish tasks,

respectively.

Equal Side with Greater

Dominant Variable

Figure la. Model of rule 1

Does . DL = DR?,

No

SL = SR?.

Side with Greater

Dominant Variable

Yes /

Equal

No

Side with Greater

Subordinate Variable

Figure lb. Model of rule 2



271

Greater Dominant Variable

Greater Subordinate

Variable

Side with Greater Guess Dominant and Greater Subordinate Variables

Figure le. Model of rule 3

Does

sDL = m.y

Yes . No

Does SL = SR?.

Does

. SL = SR?,

Yep

Equal

No Yes,

Side with Greater

Subordinate Variable

Side with Greater

Dominant Variable

No

Greater D Same Side ar .Greater S?>

Yes No

Side with Greater Use an Algorithm Dominant and Greater to Determine Side Subordinate Variables

Figure Id. Model of rule 4



272

THE BALANCE BEAM PROBLEMS

In the problems to follow there are 24 balance scales with each having equally spaced pegs along their lengths. Each balance scale would have between one and

six metal disks on one of the pegs on each side of the fulcrum. The scale is kept motionless by placing two supports under its arms.

For each problem, please answer the following question:

Which side of the scale would go down, or would it remain level (balance) when supports A and B are removed?

Please mark your answer in the box provided.

For example:

linn linn

* JX i IVVVJ ESS VM kVm

A B

In the above example, right side would go down. The answer would be:

□ left-side down

H right-side down □ remain level (balance)

Please turn the page and proceed carefully. Good luck!

Figure 2. Sample item for the balance task



273

THE MARBLES PROBLEMS

In each of the problems to follow you will be shown two piles of marbles, some

black ones and some white ones in each pile.

For each problem, please answer the following question:

Which pile of marbles would you want to choose from if you want a black marble and had to choose with your eyes closed?

Please mark your answer in the space provided.

For example:

6&£> c§J8i Pile A Eik B

In the above example, pile B would give us a better chance of picking a black marble than pile A The answer would be:

□ choose pile A IS choose pile B □ choose either one

Please turn the page and proceed carefully. Good luck!

Figure 3. Sample item for the marbles task



274

THE FISH PROBLEMS

In each of the problems to follow you will be shown two pictures of fish, picture A and picture B. Some of the fish in each picture are big and some are small. Also, some of the fish are black and others are white.

For each problem, please study the two pictures carefully and answer the

following question:

Which picture would you think has a higher relationship or correlation between the size of the fish and the color of their bodies?

Please note that a higher relationship exists if most of the big fish are black and most of the small fish are white, or, if most of the big fish are white and most of the small fish are black.

Please mark your answer in the space provided. For example:

Picture A Picture B

1X3 DO DO

In the above example, picture B shows a higher relationship between the size of

the fish and the color of their bodies than picture A. The answer would be:

□ picture A ®

picture B

□ same in both pictures

Please turn the page and proceed carefully, Good luck!

Figure 2. Sample item for the fish task



275

The rule sets associated with the three concepts differ somewhat in certain

details from each other, but essentially they all are special cases of the rules

described in Figures la-d. A difference worth mentioning is that one branch of the rule 3 strategy for the proportionality problems involves a guessing

component, whereas the same branch of the rule 3 strategy for the probability and correlation problems involves an additive algorithm. There is evidence that

additive strategies often dominate multiplicative strategies in subjects whose cog nitive schemes are less developed (Karplus, 1981; Karplus, Pulos and Stage,

1983). The rule 4 strategies for the 3 concepts all employ multiplicative algorithms.

The problem solving strategies

Figures 5-7 depict the various rules for the concepts. Figures 5a-d provide the

models for rules 1-4 for the concept of proportionality as assessed by the balance

task. In these models, the following abbreviations are used: (a) LW = the weight on the left side of the balance; (b) LD = the distance from the fulcrum of the

weight on the left side of the balance; (c) RW = the weight on the right side of the

balance; (d) RD = the distance from the fulcrum of the weight on the right side of

the balance.

' Does \

LW = RW?

Yes

Balance

No

Side with Greater

Weight Down

Figure 5a. Model of rule 1 for the concept of proportionality as assessed by the

balance task

Balance Side with Greater Distance Down

Figure 5b. Model of rule 2 for the concept of

proportionality as assessed by the balance task



276

Side with Greater Guess

Weight and Distance Down

Figure 5c. Model of rule 3 for the concept of proportionality as assessed by the balance task

Balance Side with Greater

Weight x Distance Product

Figure 5d. Model of rule 4 for the concept of proportionality as assessed by the balance task



277

Figures 6a-d present the models for rules 1^1 for the concept of probability as

assessed by the marbles task. In these models, the following conventions are

used: (a) LB = number of black marbles on the left side; (b) RB = number of

black marbles on the right side; (c) LW = number of white marbles on the left

side; and (d) RW = number of white marbles on the right side.

Equal Side with

Chances More Black

Marbles

Figure 6a. Model of rule 1 for the concept of probability as assessed by the marbles

task

Equal Side with Chances Less

White Marbles

Figure 6b. Model of rule 2 for the concept of

probability as assessed by the marbles task

Equal Side with Chances Greater

B-W Value

Figure 6c. Model of rule 3 for the concept of probability as assessed by the marbles task



278

Side with Less

White Marbles

Side with More Black

Marbles

Equal Chances

Equal Chances

Side with Greater

B/(B+W) Value

Figure 6d. Model of rule 4 for the concept of probability as assessed by the marbles task

Figures 7a - d provide the models for rules 1-4 for the concept of correlation as assessed by the fish task. In these models, the following conventions are used:

(a) LC = sum of the number of large black fish and the number of small white fish on the left side; (b) LD = sum of the number of large white fish and the num

ber of small black fish on the left side; (c) RC = sum of the number of large black fish and the number of small white fish on the right side; and (d) RD = sum of the number of large white fish and the number of small black fish on the right side.

Does LC = RC?

Yes^

Equal

No

Side with Grealer C

Value

Figure 7a. Model of rule 1 for the concept of correlation as assessed by the fish task

Side with Greater C

Value

Equal Side with Smaller D Value

Figure 7b. Model of iule 2 for the concept of

correlation as assessed by the fish task



279

Equal Side with Greater C-D Value

Figure 7c. Model of rule 3 for the concept of correlation as assessed by the fish task

Equal Side with Greater

l(C - D)/(C+D)I Value

Figure 7d. Model of rule 4 for the concept of correlation as assessed by the fish task



280

Transition to higher rules by direct instruction

Inhelder and Piaget (1958) conjectured that one must pass through a phase of cog nitive disequilibrium prior to attainment of the period of formal operations. In this

phase of cognitive disequilibrium, individuals recognize the inadequacies of prob lem solving strategies that are appropriate to concrete operational thought and

seek more powerful problem solving strategies that are appropriate to formal

operational thought. Rule 4 for each concept assessed in this study entails formal

operational reasoning. Siegler found that cognitive transitions between rule 1 and

rule 2 and between rule 2 and rule 3 for the concept of proportionality, for exam

ple, could be promoted by providing corrective feedback with the test items and/ or by training subjects to encode the subordinate variable.

This study attempted to establish cognitive disequilibrium to promote transition

to rule 4 for each concept. Whereas in Siegler's research, the subjects were chil

dren and adolescents, in this study they were adult teachers. The method used was

"direct instruction", an expository approach wherein teachers were provided an

explanation for inadequacies in their cognitive rules and were given corrective

feedback information (rules and procedures) for rule 4 solution of the test items.

This expository approach also focused on worked or "best" examples of the pro

portion, probability correlation problems. The exact sequence of instruction with

examples is described in more detail below.

Some evidence exists that the direct teaching of a best example or "exemplar model" of a concept can promote its acquisition (Tennyson et al., 1983; Rosch,

1975, 1978; Smith and Medin, 1981). If a best example of a concept is taught in

conjunction with strategy information for the rules and procedures for concept

identification, then concept learning is enhanced even more (Tennyson et al.,

1975). For example, Petkovich (1986) showed that direct teaching of worked

algebra problem examples, together with strategy information, can promote the

acquisition of underlying concepts. This study explores the feasibility of using direct instruction as a means of

facilitating transition through a phase of cognitive disequilibrium to the period of formal operational thought. The general question is: Can direct instruction be used as a method to effect transition in teachers from lower to higher cognitive rules?

Method

Subjects

Twenty eight teachers of grades four, five and six enrolled in a four week

(32 hour) mathematics education workshop on problem solving. Seven teachers were male, and 21 were female. All teachers were currently teaching in the sub urbs of a large Minnesota city.



281

Procedure

The teachers were administered three cognitive pretests at the start of the work

shop and three cognitive posttests at the end of the workshop. Midway through

the workshop, or during the fourth class meeting, there was a four hour session on

Siegler's rule assessment approach and the cognitive rules used to solve the

Awang cognitive test items.

Workshop instruction

The session on Siegler's rule assessment approach and the cognitive rules to solve the Awang cognitive test items was placed in the context of teaching teachers about cognitive development. Teachers were told that the Siegler and Awang

work could help them understand how children might think about fundamental

mathematical concepts and also help them, as teachers, think through these ideas. An expository or direct instruction method, combined with strategy informa

tion and several worked examples, was used during the four hour session. Teachers were first given a copy of the balance beam pretest and asked to mark

their answers. After completing the test, the variables of weight and distance were

identified as fundamental variables for the balance beam task. A cognitive rule for the balance beam task was defined as a strategy for deciding whether the bal ance beam would tip to the left, right, or balance, based on the values of the

weight and distance variables.

Teachers were next given a worksheet on which the left (L) and right (R) val

ues for the variables weight (W) and distance (D) were recorded for each item.

Figure 8 shows a copy of this worksheet. Item #1 has a left weight (LW) equal 2, left distance (LD) equal 4, right weight (RW) equal 4, and right distance (RD)

equal 4. Teachers were then told that an examinee who follows rule 1 only attends to the dominant variable of weight. They were then asked to mark, on the rule I column of their worksheet, how an examinee following a rule 1 strategy

would respond for each item.

The procedure for rule 1 was then repeated for rules 2-4. Rule 2 was explained

as a strategy wherein the examinee attends to the subordinate variable, distance,

only if the values for the dominant variable, weight, are equal; otherwise, the

examinee follows rule 1. Rule 3 was explained as a strategy that expands on

rule 2 in which the examinee attends to dominant and subordinate variables for all

cases. In particular, rule 3 helps settle the case where the values of the dominant

variable are not equal but the values of the subordinate variable may or may not

be equal. Teachers were given a copy of the flowchart for rule 3 (similar to

Figure 5c) to help them think through this rule. Rule 4 was explained as a strategy that expands upon rule 3 by replacing the "guessing" or "muddling through" with



282

an algorithm wherein distance is multiplied by weight to determine the side with the greatest torque.

For each of the rules 1-4, teachers were asked to respond in the corresponding column how an examinee following that rule would respond for each item (see Figure 8). Thus, teachers had five responses for each item: the values for thinking "as if' they were following rule 1 - rule 4 and their original response for the item before instruction. Based on these data, teachers were then asked to determine which cognitive rule they were most likely following. This activity generated some dissonance and disequilibrium as teachers discussed among themselves which rules they were following and also thought about the possibility of using more sophisticated rules.

The procedure mentioned heretofore of taking the test, identifying dominant

and subordinate variables, determining rule 1 - rule 4 responses, and working through the items "as if' an examinee were following each of the rules, was

repeated for the probability and the correlation pretests.

Expected Answers: Balance Beam Problems

1

2

3

4

5

6

7

8

9

10

11

12

Values Rules Your Answer

LW LD RW RD I n m IV

2 4 4 4

4 4 6 2

3 113

3 2 3 4

4 2 4 2

4 12 4

4 4 6 4

3 4 4 3

4 4 4 3

3 4 14

3 4 5 2

5 2 5 4

Figure 8. Sample worksheet used by teachers during the Direct Instruction session



283

Pre- and posttests

The pre- and posttests were constructed by subdividing each of the three Awang tests into two subtests. The odd items in an original Awang test constituted one

subtest of 12 items and the even items constituted another subtest of 12 items.

There were thus six resulting tests: three pretests and three posttests.

The response patterns for each of the six tests were analyzed in terms of the

extent to which they comply with the response patterns produced by each of the

four cognitive rules determined for each concept. Teachers were classified as

complying with a given cognitive rule for a specific test if their response patterns most closely fitted the response pattern generated by that cognitive rule when

compared to the response pattern generated by the other three cognitive rules for

the test.

The classification scheme can be best explained by thinking of a teacher's

score on each of the 4 rules as a vector in 4-space. The four components of this

vector are the teacher's scores on each of the four rules. The score of a teacher

who followed a "pure rule 1" might also be thought of as a vector in 4-space. The

components of the pure rule 1 vector are the scores of a teacher on each of the

four rules, under the assumption that he/she followed a pure rule 1 strategy.

Similarly, there are pure rule 2, pure rule 3, and pure rule 4 vectors. A teacher

was classified by computing the Euclidean distance between his/her score and

each of these four pure rule vectors and selecting that rule whose pure rule vector

was the closest to the teacher's score vector. This might be compared with

Pearson's "least squares criterion", which is used to find the regression line which

best fits a set of data points (cf., Minium, 1978, p. 179).

Unfortunately, the distances between successive pure rule vectors are not

equal. This implies that the rule assignments do not constitute an interval level of

measurement. When testing hypotheses about differences between rule

assignments due to the direct instruction treatment, it was thus necessary to use

nonparametric methods (cf., Siegel, 1956).

Results

One major finding was that not all the elementary school teachers manifested the

highest level of cognitive development for each of the three concepts. There was

substantial variation regarding the cognitive rules being manifested by the teach

ers for each of the three pretests. To be specific, for the balance beam proportion

pretest, 2 teachers were classified as using rule 2, 9 as using rule 3, and 17 as

using rule 4. On the marbles probability pretest, 2 teachers were classified as

using rule 3, and 26 as using rule 4. On the correlation pretest involving "fish"

items, 3 teachers were classified as using rule 1, 1 as using rule 2, 12 as using



284

rule 3, and 12 as using rule 4. Thus, there were substantial numbers of teachers

who were either using defective strategies in solving the items or were seemingly inconsistent in the strategy they did use or simply guessed.

As a result of the workshop, there were indicated in the posttest data some

significant gains. On both the balance beam proportion posttest and the marbles

probability posttest, all teachers were classified as using rule 4. On the correlation

posttest involving "fish" items, 2 teachers were classified as using rule 1, 1 as

using rule 2, 13 as using rule 3, and 12 as using rule 4. These shifts between pre and posttests are indicated in Tables 1-3.

Due to the fact that the rules do not constitute an interval scale of measure

ment, the non-parametric test, the sign test, was used to assess the hypothesis that

the probability of a posttest rule placement exceeding the pretest rule placement

for any given subject is .50 (cf., Siegel, 1956). The results are found in Table 4.

For the balance task, the sign test engendered 11 positive shifts out of a total of

11 nonzero shifts, which engendered a p-value of .00. Thus, the posttest rules of

the subjects were significantly greater than the pretest rules. For the marbles task, the sign test engendered 2 positive shifts out of a total of 2 nonzero shifts, which

engendered a p-value of .25. This shift was not significant. For the "fish" task, the

sign test engendered 9 positive shifts and 7 negative shifts out a total of 16 non zero shifts, which engendered a p-value of .40. This improvement was not

significant.

Discussion

The data support the contention that the mathematics in-service workshop, with

its four hour session on Siegler's rule assessment research, results in significant

improvements in the levels of cognitive development regarding the cognitive

scheme associated with the concept of proportion. Though the workshop did not

result in significant improvements in the probability concept, all the teachers were

assigned to rule 4 on the probability posttest. The workshop did not result in a sig nificant improvement, on the average, in the levels of cognitive development regarding the cognitive scheme associated with the concept of correlation.

One reason for the inability of over 50% of the teachers to attain rule 4 for the concept of correlation is the basic complexity of the algorithm in rule 4. The rule 4 algorithm for the concept of proportion involves two encoded variables,

weight and distance, and one procedural step, their product. The rule 4 algorithm for the concept of probability involves two encoded variables and two procedural

steps. The rule 4 algorithm for the concept of correlation involves four encoded variables and seven procedural steps. The teachers who were not able to attain

rule 4 for the concept of correlation may not have had the necessary cognitive capabilities to do so - e.g., inadequate chunking strategies and woiking memory capacities to chunk and encode the necessary variables.



285

Table 1. Frequencies of subjects manifesting different cognitive rules for the proportionality concept for the pre- and posttests

Pretest rules Posttest rules

1

2

3

4

0

0

0

0

0

0

0

0

0

0

0

0

0

2

9

17

Table 2. Frequencies of subjects manifesting different cognitive rules for the probability concept for

the pre- and posttests

Pretest rules Posttest rules

12 3 4

1 0 0 0 0

2 0 0 0 0

3 0 0 0 2

4 0 0 0 26

Table 3. Frequencies of subjects manifesting different cognitive rules for the correlation concept for

the pre- and posttests

Pretest rales Posttest rules

1 2 3 4

10 0 12

2 0 0 1 0

3 0 0 7 5

4 2 14 5



286

Table 4. Sign test of hypotheses that the probability of a posttest rale placement exceeds the pretest rale placement

Posttest/pretest Concept Rule placement Shift Proportionality Probability Correlation

Positive 11 2 9

Negative 0 0 7

Neither 17 26 12

p-value .00 .25 .40

At least three limitations of the above study need to be conceded. The first is

the "ceiling" effect noted for the proportionality and the probability tasks when

used with adult teachers. The tests were originally designed for younger children, and it was to be expected that many adults would likely score very high on the

measures of proportionality, probability and correlations. Nonetheless, even with

the adult population, there are still some significant variation and shifts in the rule

usage as a result of the workshop intervention. More challenging tests of propor

tionality and probability might result in the ability to assess more accurately transition between rule 3 and rule 4. The correlation test, on the other hand,

showed no ceiling effect, perhaps due to the complexity of the rule for this

concept.

A second limitation pertains to the link between the concepts of proportional

ity, probability and correlation and the Awang pencil-and-paper tests. In their

original investigation of proportionality, probability and correlation, Inhelder and

Piaget (1958) used clinical interviews, which allow for greater variation in subject

response. It might be argued that Inhelder and Piaget's notions of proportionality,

probability, and correlation are much richer than those used in this study. Perhaps the Awang tests do not tap these concepts fully. This limitation, which might be

regarded as one affecting the construct validity of the tests, must be considered.

However, even though the Awang tests may not capture the richness of Inhelder

and Piaget's original work, they do provide objective and reliable measures of

formal reasoning conceptual ability. One might regard the Awang tests as provid

ing operational definitions for the concepts of proportionality, probability, and

correlation.

A third limitation pertains to the conjectural status of Piaget and Inhelder's and

Siegler's theories. This study has not included a full description of the evidence,

pro and con, for either of these theories. A complete evaluation of the concepts of

proportionality, probability and correlation would need to say more about this

evidence and competing theories which might explain the same phenomena.



287

On the other hand, the present study does permit, in a sense, a more scientific

examination of the original views of Inhelder and Piaget. The original views

regarding proportionality, probability and correlation were proposed against the

backdrop of Piaget's theory of logical operations. In fact, Inhelder and Piaget

(1958) interpreted levels of understanding of those three concepts in terms of the

logical operational structures which characterize the various cognitive stages within Piagetian cognitive theory.

Though very rich, Piaget's theory of logical operations is difficult to test with

out the extensive clinical interviews done by Inhelder and Piaget. However, the

Awang tests, which permit a precise examination of what might be regarded as

operational definitions of proportionality, probability and correlation, permit the

formulation of hypotheses about these concepts which are falsifiable. This

criterion of "falsifiability" is regarded by philosophers of science as one of the

hallmarks of scientific theorizing (Popper, 1963). Furthermore, the Awang tests

have proved to be useful in classifying the cognitive states of the workshop teach

ers with respect to these concepts of proportionality, probability, and correlation.

This paper indicates, to some degree, that the Siegler rule-based assessment

method influenced by the work of Inhelder and Piaget can be of use in assessing the cognitive effects of direct instruction with adult teachers. This method thus

manifests "fruitfulness" (or explanatory power), which is regarded by some phi

losophers of science as a further hallmark of scientific theorizing (Lakatos, 1970). The concepts of proportionality, probability and correlation are likely prerequi

sites for achievement in disciplines such as physics, chemistry, statistics, and

other secondary and post-secondary subject matter areas in mathematics,

engineering and the physical sciences. These pivotal concepts of proportionality,

probability and correlation are central concepts in formal operational reasoning,

viewed by developmental psychologists as constituting the most advanced stage of cognitive development (Inhelder and Piaget, 1958). This stage of formal opera tions marked by the concepts of proportionality, probability and correlation is

attained by only approximately 30% of the adolescents and adults in the U.S.A.

(e.g., McKinnon and Renner, 1971; Schwebel, 1975). Perhaps there are biologi

cally-imposed limits as to what percentage of adults can become formal reasoners

across a variety of advanced concepts regardless of the intervention techniques that are employed - e.g., direct instruction of cognitive rules.

This study indicates that mathematics in-service workshops can be fashioned

in such a way as to result in significant positive cognitive changes among

elementary school teachers. Cognitive developmental research, coupled with

contemporary advances in cognitive assessment, can be used effectively in the

construction of workshop sessions to deepen and enrich the mathematical under

standings of elementary school teachers. This study also exemplifies how the rule

assessment approach, which is presently so important in the area of cognitive

development, can be successfully applied to study cognitive change among teachers resulting from teacher education projects.



288

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The cognitive effects of a mathematics in-service workshop on elementary school teachers